tppreqb - Metamath Proof Explorer

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Theorem tppreqb 4104

Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)

**Assertion**
Ref	Expression
tppreqb	$/\$ $/\$

**Proof of Theorem tppreqb**
Step	Hyp	Ref	Expression
1		3ianor 1024	. . . 4 $/\$ $/\$ $\/$ $\/$
2		df-3or 1008	. . . 4 $\/$ $\/$ $\/$ $\/$
3	1, 2	bitri 257	. . 3 $/\$ $/\$ $\/$ $\/$
4		orass 533	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
5		ianor 496	. . . . . . . 8 $/\$ $\/$
6		tpprceq3 4103	. . . . . . . 8 $/\$
7	5, 6	sylbir 218	. . . . . . 7 $\/$
8		tpcoma 4059	. . . . . . 7
9		prcom 4041	. . . . . . 7
10	7, 8, 9	3eqtr3g 2528	. . . . . 6 $\/$
11		orcom 394	. . . . . . . 8 $\/$ $\/$
12		ianor 496	. . . . . . . 8 $/\$ $\/$
13	11, 12	bitr4i 260	. . . . . . 7 $\/$ $/\$
14		tpprceq3 4103	. . . . . . 7 $/\$
15	13, 14	sylbi 200	. . . . . 6 $\/$
16	10, 15	jaoi 386	. . . . 5 $\/$ $\/$ $\/$
17	4, 16	sylbi 200	. . . 4 $\/$ $\/$ $\/$
18	17	orcs 401	. . 3 $\/$ $\/$
19	3, 18	sylbi 200	. 2 $/\$ $/\$
20		df-tp 3964	. . . 4
21	20	eqeq1i 2476	. . 3
22		ssequn2 3598	. . . 4
23		snssg 4096	. . . . . . 7
24		elpri 3976	. . . . . . . 8 $\/$
25		nne 2647	. . . . . . . . . 10
26		3mix2 1200	. . . . . . . . . 10 $\/$ $\/$
27	25, 26	sylbir 218	. . . . . . . . 9 $\/$ $\/$
28		nne 2647	. . . . . . . . . 10
29		3mix3 1201	. . . . . . . . . 10 $\/$ $\/$
30	28, 29	sylbir 218	. . . . . . . . 9 $\/$ $\/$
31	27, 30	jaoi 386	. . . . . . . 8 $\/$ $\/$ $\/$
32	24, 31	syl 17	. . . . . . 7 $\/$ $\/$
33	23, 32	syl6bir 237	. . . . . 6 $\/$ $\/$
34		3mix1 1199	. . . . . . 7 $\/$ $\/$
35	34	a1d 25	. . . . . 6 $\/$ $\/$
36	33, 35	pm2.61i 169	. . . . 5 $\/$ $\/$
37	36, 1	sylibr 217	. . . 4 $/\$ $/\$
38	22, 37	sylbir 218	. . 3 $/\$ $/\$
39	21, 38	sylbi 200	. 2 $/\$ $/\$
40	19, 39	impbii 192	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $\/$ wo 375 $/\$ wa 376 $\/$ w3o 1006 $/\$ w3a 1007

wceq 1452

wcel 1904

wne 2641

cvv 3031

cun 3388

wss 3390

csn 3959

cpr 3961

ctp 3963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451

This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-sn 3960 df-pr 3962 df-tp 3964

This theorem is referenced by: (None)

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